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Theorem rexlimdva2 39653
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
rexlimdva2.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
rexlimdva2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Distinct variable groups:   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimdva2
StepHypRef Expression
1 rexlimdva2.1 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
21exp31 629 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
32rexlimdv 3059 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ral 2946  df-rex 2947
This theorem is referenced by:  supminfxr  40007  infrpgernmpt  40008  limsupresxr  40316  liminfresxr  40317  liminflelimsuplem  40325  limsupgtlem  40327  liminfvalxr  40333  liminfreuzlem  40352  cnrefiisplem  40373  xlimmnfvlem2  40377  xlimpnfvlem2  40381  smfliminflem  41357
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